8. \( x^{2}+2 x+y^{2}+2 y=59 \)

This equation represents a circle in the \(xy\)-plane. If we rearrange it, we can complete the square for both the \(x\) and \(y\) terms. This leads us to \( (x+1)^{2} + (y+1)^{2} = 59 - 2 - 2 = 55 \). With its center at \((-1, -1)\) and a radius of \(\sqrt{55}\), it gives us a neat geometric visualization! Moreover, you can see how such equations are useful in various fields. For instance, in physics, circular motion is often described using equations of circles. Knowing how to analyze such equations can help one solve complex problems in kinematics and even in video game design where movement can be modeled using circular paths.